the transmission is one at every location except when the magnitude is
zero. When the magnitude is zero, both components are set to zero.
The Fourier transform of a test square is shown in Figure 7.2. The
same transform with high-frequency emphasized is shown in Figure 7.3.
Note in Figure 4.1 that the edges are sharply accentuated in the
modified image while broad areas are dark. Despite the modified
frequency, the image is still easily recognized. The Fourier
transform of a phase-only-filtered image has minimum dynamic range,
but the image is distorted by the extreme edge enhancement typical of
phase-only filtering. When all pre-emphasis is completed, the
reference and test patterns are normalized to possess continuous
values between 0 and 1 inclusive. Normalization is performed to
simulate the action of a transparency as would be used in an optical
correlator and allows the determination of light efficiency.
Once the reference transform is modified according to the desired
pre-processing technique, and the point by point product taken with
the test image transform, the result is transformed again. Note that
the inverse is not taken because the lenses which are simulated can
only take forward Fourier transforms. Recall that the only difference
between a forward and inverse transform is that the result will be
inverted and perverted (see equation 2.11). It is of no consequence
in this case that the correlation is upside-down. Figures 7.4 through
7.6 show the auto-correlation of a square with no pre-emphasis,
high-frequency emphasis, and phase-only filtering applied. Note that
the correlation spikes with pre-emphasis are considerably sharper.
This is to be expected as the correlation length is inversely related
to the high-frequency content of the images.